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Econ-634 · economics · Economics

Biases under high stakes and experience

LLM status Open unverified

Statement

Original subject group: Behavioral and experimental economics

Audit classification: Background; proposed extension. The genuine 2023 article already tests large incentives in four biases. Professional expertise, repeated feedback and matched market transport are editorial extensions, not an explicitly verified remaining combined agenda.

Definitions and precise research target

Model and research target. Fix a decision task $k$ with objectively correct response $y_k^*(X)$ for problem attributes $X$. Let $Y_{ik}(s,t)$ be individual $i$'s response when assigned incentive level $s\geq0$ after $t$ opportunities to practice with feedback. Define a preregistered bias score $B_{ik}(s,t)=b_k(Y_{ik}(s,t),y_k^*(X),X)$, where $b_k$ distinguishes a systematic directional error from unsystematic inaccuracy. Let $H_i$ denote observed professional expertise and $P$ a specified target population. The relevant response surface and incentive contrast are \[\mu_{k,P}(s,t,h)=\mathbb E_P[B_{ik}(s,t)\mid H_i=h],\qquad \tau_{k,P}=\mu_{k,P}(s_H,t,h)-\mu_{k,P}(s_L,t,h).\] Estimate these quantities using randomized incentives and feedback, adequate support for both stake levels, and payments expressed relative to participants' economic resources. Expertise is not automatically randomized; comparisons across $h$ require an explicit selection or causal design. Which tasks retain materially positive bias at high stakes after meaningful experience, and how accurately do laboratory estimates predict bias in matched consequential market decisions? Specify thresholds for economic importance and distinguish learning from selective attrition, fatigue, and changes in task interpretation.

Definitions and maintained assumptions (editorial unless a source definition is named). A task $k$ supplies attributes $X$, an objectively correct answer $y_k^*(X)$ and a prespecified directional-error score; decisions reflecting legitimate taste differences cannot be assigned a correct answer without a normative model. Normalize incentives by a declared income/resource measure and specify the payment scoring rule: increased nominal maximum prizes do not uniquely define increased marginal incentives. Potential response $Y_{ik}(s,t)$ refers to assigned stake $s$ and a controlled training/feedback protocol of length $t$; experience is an intervention, not elapsed time alone. Expertise $H_i$ is baseline and may be selected. Define a threshold $b_0>0$ for material residual bias and contrasts $\mu(s_H,t,h)-\mu(s_L,t,h)$ only on expertise support. Randomized stakes/feedback identify effects for the sampled population, accounting for repeated measures and attrition; causal expertise effects require another design. Field prediction must preserve task attributes, score meaning and comparable financial consequences. The source’s four task results cannot settle this entire surface.

Variants and assumption changes

  1. Feedback interaction (editorial). Randomize stakes and the number/type of feedback opportunities factorially. Estimate whether high stakes change learning speed and residual directional bias.
  2. Expertise transport (editorial). Compare a prespecified professional population with matched nonprofessionals under identical tasks, and test forecasts on consequential matched decisions. Report predictive differences separately from causal effects of acquiring expertise.

References and source audit

  1. Introduction p.818; discussion p.830. Supports the stated research area; exact displayed specification is editorial. https://benjamin-enke.com/pdf/Biases_stakes.pdf
  1. Benjamin Enke; Uri Gneezy; Brian Hall; David Martin; Vadim Nelidov; Theo Offerman; Jeroen van de Ven. 2023. Cognitive Biases: Mistakes or Missing Stakes?. §I, p. 818; §IV Discussion, p. 830 (PDF pp. 1 and 13).. https://benjamin-enke.com/pdf/Biases_stakes.pdf. DOI: https://doi.org/10.1162/rest_a_01093.

Common conventions: Source mathematical conventions

These conventions apply to each entry unless explicitly replaced.

Models and identification. A primitive model $m$ specifies all probability laws, feasible choices, information, equilibrium and measurement rules used by its target. The admissible class $\mathcal M$ is fixed independently of outcomes. If $P_O(m)$ is its observable probability law and $\tau(m)$ the target, its identified set at observable law $P$ is \[ \mathcal I_\tau(P)=\{\tau(m):m\in\mathcal M,\ P_O(m)=P\}. \] Point identification means that this set is a singleton. An empty set signals incompatibility of data and assumptions. Coordinate bounds need not describe the joint set. Bounds are sharp only if every retained target is attainable or arbitrarily approachable by compatible models. Finite bounds require explicit restrictions on unobserved quantities; unrestricted confounding, hidden wealth, extrapolation or arbitrary equilibrium selection can yield uninformative sets.

Causal interventions. A potential outcome $Y_i(p)$ is the outcome under a complete policy regime $p$, including timing, financing, information and market spillovers. The target population and units are fixed before treatment. With interference, use the complete assignment vector or a justified exposure mapping; individual no-interference notation alone cannot identify an economy-wide intervention. A difference of mean potential outcomes does not require the cross-world joint distribution. Conditioning on post-treatment migration, survival, enrollment or employment changes the estimand and needs separate justification.

Statistical statements. An estimator, confidence set, forecast or test specifies a sampling law, model class and loss/coverage criterion. Uniform $(1-\alpha)$-coverage means $\inf_{m\in\mathcal M}\Pr_m\{\tau(m)\in C_n\}\geq1-\alpha$, or an explicitly asymptotic version. The whole parameter space trivially has coverage, so informativeness requires a stated width, risk or power objective. A forecasting improvement is relative to a named benchmark, information set and evaluation population.

Equilibrium and optimization. An equilibrium comprises feasible plans, optimality, beliefs where needed, budget constraints and all market/resource clearing. The primitives or a stated benchmark define each correspondence $\mathcal E(p;m)$. Nonemptiness is assumed or proved, never inferred just from compact policy sets. A selected-equilibrium target uses a declared selection rule; a robust target quantifies over all permitted equilibria. Use suprema or infima unless attainment is established. An $\varepsilon$-optimal policy is feasible and within $\varepsilon$ of the finite optimum value. Report an empty feasible set separately.

Welfare and accounting. Normative utility, interpersonal normalization, social weights, numeraire, discounting and the use of public receipts are primitives. Behavioral utility need not coincide with the welfare criterion. Household welfare includes ownership income and rents once; adding profits again to compensating variation generally double-counts them. A monetary equivalent is defined by an explicit transfer and price convention, with monotonicity and range conditions. Average effects, mechanism contributions, transfers and welfare losses are different targets. Infinite sums require convergence and dynamic feasible plans require terminal or no-Ponzi conditions.

Source versions. A working-paper date, revision date and journal publication year are distinguished. A DOI for a journal version is not automatically the DOI of an earlier manuscript. Locators refer to the stated version and printed pages where specified. A metadata-only check does not verify a claim about a full-text conclusion. Every entry's source subsection narrows the provenance claim accordingly.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-634-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-634

High stakes and experience: effort, persistent mental models, and material bias

Status: Unresolved. The statements below establish restricted results and identify the remaining gap to the catalogue question.

Question and scope

Which systematic errors survive high incentives and meaningful feedback? This attempt solves a model in which effort improves execution of an imperfect mental model. It derives a material-bias threshold and a change in the interaction between training and stakes. It then shows what selective attrition permits one to conclude. The model explains possible patterns; it does not establish the prevalence of those patterns among professionals or in markets.

An objective answer and a subjective target

Normalize the objectively correct answer to zero. A preregistered directional bias score is the response itself, oriented so positive values represent the error of interest. Let $a>0$ be an initial anchor. The participant's internal model treats $m\in[0,a)$ as correct. Effort $e\in[0,1]$ produces response \[ Y=(1-e)a+em+\varepsilon, \] where ${\mathbb{E}}\varepsilon=0$, ${\operatorname{Var}}(\varepsilon)<\infty$, and its distribution does not depend on effort or stakes. The subject is risk neutral in the relevant reward units and minimizes subjective expected quadratic scoring loss plus effort cost, \[ s{\mathbb{E}}[(Y-m)^2]+\frac c2e^2, \] where $s\geq0$ is the marginal stake measured relative to a fixed personal resource scale and $c>0$ is effort cost in the same units. Actual scoring is based on the true answer; the objective above assumes the participant predicts that answer using $m$. This is an explicit misspecification, not correct-belief maximization.

Feedback is delivered through a controlled protocol. The internal representation after $t$ rounds is assumed to follow \[ m_t=m_\infty+(1-\lambda)^t(m_0-m_\infty),\qquad 0\leq m_\infty\leq m_0<a,\quad 0<\lambda<1. \] This can result from the update $m_{t+1}=(1-\lambda)m_t+\lambda m_\infty$: objectively correct feedback is persistently encoded with distortion $m_\infty$. The model therefore distinguishes opportunities for feedback from successful interpretation of it. Setting $m_\infty=0$ removes persistent distortion.

Theorem (Effort and residual directional bias): For any $m\in[0,a)$ the unique optimal effort and mean bias are \[ e^*(s,m)=\frac{2s(a-m)^2}{c+2s(a-m)^2},\qquad \mu(s,m)=m+\frac{c(a-m)}{c+2s(a-m)^2}.\tag{1} \] For fixed $m$, bias strictly decreases with $s$, and $\lim_{s\to\infty}\mu(s,m)=m$. At positive stakes, lowering $m$ strictly lowers bias and raises effort. Consequently \[ \lim_{t\to\infty}\lim_{s\to\infty}\mu(s,m_t) =\lim_{s\to\infty}\lim_{t\to\infty}\mu(s,m_t)=m_\infty. \]
Proof: Put $d=a-m>0$. Subjective loss equals $s d^2(1-e)^2+s{\operatorname{Var}}(\varepsilon)+ce^2/2$, a strictly convex quadratic. Its first-order solution in (1) lies in $[0,1)$ and is therefore the unique constrained optimum. Since ${\mathbb{E}} Y=a-de$, substitution gives the bias formula. Differentiation gives \[ \mu_s=-\frac{2cd^3}{(c+2sd^2)^2}<0, \] \[ \mu_m=\frac{6csd^2+4s^2d^4}{(c+2sd^2)^2}>0\quad(s>0). \] Effort increases with $d$ at positive stakes, so it rises when $m$ falls. The limits follow from (1), continuity in $m$ and the explicit convergence $m_t\to m_\infty$. ∎

The mean score isolates systematic direction. Increasing zero-mean response noise would raise mean squared error without changing $\mu$; treating that change as stronger directional bias would confuse two outcomes. Similarly, effort can rise markedly while the biased target remains. An increased response time is not by itself evidence that the mental model has become correct.

A threshold for economically material bias

Fix a declared materiality threshold $b_0\in(0,a)$. The threshold is part of the research design; it cannot be chosen after observing which comparison happens to be statistically significant.
Proposition (When finite stakes can cross the threshold): At a fixed training level $t$, if $m_t\geq b_0$, no finite stake achieves $\mu(s,m_t)\leq b_0$. If $m_t<b_0$, the condition is equivalent to \[ s\geq\frac{c(a-b_0)}{2(a-m_t)^2(b_0-m_t)}.\tag{2} \] In particular, if $m_\infty\geq b_0$, no combination of finite training and finite stakes eliminates material bias in this model. If $m_\infty<b_0$, sufficiently much training followed by a sufficiently large finite stake does.
Proof: For finite $s$, (1) is strictly greater than $m_t$ because $a-m_t>0$. This proves the first case. In the second, subtract $m_t$ from both sides of $\mu\leq b_0$ and multiply by the positive denominator. Rearrangement gives $c(a-b_0)\leq2s(a-m_t)^2(b_0-m_t)$, proving (2). If the limiting mental-model error is below the threshold, convergence of $m_t$ ensures some finite $t$ with $m_t<b_0$; then the right side of (2) is finite. The opposite case follows from $m_t\geq m_\infty$. ∎

The incentive--training interaction changes sign

Treat a small training improvement as a reduction in the scalar $m$, keeping the anchor and effort-cost technology fixed. This local comparative static does not assert that every real training program changes only $m$.
Proposition (A sign-changing local interaction): The mixed derivative of mean bias is \[ \mu_{sm}=\frac{2cd^2(3c-2sd^2)}{(c+2sd^2)^3},\qquad d=a-m. \] Therefore lowering $m$ makes the negative stake derivative more negative when $2sd^2<3c$, but less negative when $2sd^2>3c$. Training and stakes need not have a single interaction sign even in this one-parameter mental-model family.
Proof: Differentiate $\mu_s=-2cd^3/(c+2sd^2)^2$ with respect to $d$, and multiply by $\partial d/\partial m=-1$. This gives the displayed derivative. A small improvement has $dm<0$, so the change in $\mu_s$ has the opposite sign to $\mu_{sm}$. The two inequalities give the stated conclusions; equality gives zero local interaction. ∎

This result distinguishes a reduction in the level of bias, which always follows a reduction in $m$ at positive stakes, from an increase in the marginal effectiveness of incentives, whose sign varies. A factorial experiment can measure the latter without treating expertise as randomly assigned.

Attrition and a claim of persistent bias

For an empirical task, rescale the preregistered directional score to lie in $[-1,1]$. In one randomized stake/training cell let $R$ indicate an observed response, $\pi=\Pr(R=1)$, and $\mu_{\mathrm{obs}}={\mathbb{E}}[B\mid R=1]$ when $\pi>0$. Target the original assigned population, not the selected respondents. Make no restriction on why participants disappear.
Proposition (Sharp attrition bounds): For $\pi>0$, the full assigned-population mean is in the sharp interval \[ [\pi\mu_{\mathrm{obs}}-(1-\pi),\quad \pi\mu_{\mathrm{obs}}+(1-\pi)].\tag{3} \] For $\pi=0$ the sharp interval is $[-1,1]$. Thus a population claim that residual mean bias exceeds $b_0$ is robust to arbitrary attrition only when the lower endpoint exceeds $b_0$.
Proof: Decompose the mean as $\pi\mu_{\mathrm{obs}}+(1-\pi){\mathbb{E}}[B\mid R=0]$. The unobserved conditional mean lies in $[-1,1]$, giving the bounds. Any point in that interval is realized by assigning every missing response the corresponding constant score, without altering the observed response law or observation probability. This proves sharpness, including the fully missing case. The robustness criterion follows immediately from the identified set. ∎

For two treatment cells with no cross-cell restrictions, the sharp contrast interval is obtained by subtracting the upper endpoint of one cell from the lower of the other, and vice versa. Sampling uncertainty in all endpoints still requires valid inference; population bounds are not confidence intervals. Selective retention of experienced participants can otherwise imitate either learning or persistent bias.

Completion Estimate

51%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt derives incentive and training responses in an imperfect-mental-model framework, characterizes material residual bias and changing interaction signs, and proves sharp retained-cohort attrition bounds. The full response-surface target still requires randomized stakes and feedback with measured expertise, validated task meaning, normalized payment incentives, uncertainty and attrition handling, and accurate prediction of matched consequential market decisions.

Remaining gap and reference

The effort model assumes risk neutrality in normalized payments, quadratic subjective loss, a fixed anchor and a specified feedback-encoding law. It does not identify those primitives or show that they transport to professionals. A full solution needs randomized stakes and feedback, stable task meaning, a baseline expertise definition, retained-cohort outcomes, and out-of-sample prediction for matched consequential decisions. Existing high-stakes experiments address part of that agenda; this attempt does not relabel their findings as unproved.

Benjamin Enke, Uri Gneezy, Brian Hall, David Martin, Vadim Nelidov, Theo Offerman and Jeroen van de Ven (2023), Cognitive Biases: Mistakes or Missing Stakes?, Review of Economics and Statistics 105(4), 818--832, especially pp. 818 and 830, https://doi.org/10.1162/rest_a_01093; author manuscript https://benjamin-enke.com/pdf/Biases_stakes.pdf. The source motivates the incentive question; the mental-model calculations and attrition analysis here are the present restricted exercise.

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